# Meantone family

(Redirected from Flattone)

The 5-limit parent comma of the meantone family is the Didymus or syntonic comma, 81/80. This is the one they all temper out. The monzo for 81/80 goes |-4 4 -1>, and that can be flipped around to the corresponding wedgie, <<1 4 4||, which tells us that the period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.

# 5-limit meantone

Comma: 81/80

POTE generator: ~3/2 = 696.239

Mapping generator: ~3

valid range: [685.714, 720.000] (7 to 5)

nice range: [694.786, 701.955] (1/3 comma to Pythagorean)

strict range: [694.786, 701.955]

Map: [<1 0 -4|, <0 1 4|]

EDOs: 5, 7, 12, 19, 26, 31, 43, 45, 50, 55, 67, 69, 74, 81, 88, 98, 105, 117, 131b, 212bb, 293bb

## Seven limit children

The 7-limit children of 81/80 are septimal meantone, with normal comma list [|-4 4 -1>, |-13 10 0 -1>], flattone, with normal list [|-4 4 -1>, |-17 9 0 1>], dominant, with normal list [|-4 4 -1>, |6 -2 0 -1>], sharptone, with normal list [|-4 4 -1>, |2 -3 0 1>], injera, with normal list [|-4 4 -1>, |-7 8 0 -2>], mohajira, with normal list [|-4 4 -1>, |-23 11 0 2>], godzilla, with normal list [|-4 4 -1>, |-4 -1 0 2>], mothra, with normal list [|-4 4 -1>, |-10 1 0 3>], squares, with normal list [|-4 4 -1>, |-3 9 0 -4>], and liese, with normal list [|-4 4 -1>, |-9 11 0 -3>].

# Septimal meantone

The comma |-13 10 0 -1> for septimal meantone tells us that the interval class for 7 is 10 generator steps up. Hence, the 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, and 7/5, C-F#, the tritone. The wedgie for septimal meantone is <<1 4 10 4 13 12||, again telling us how to get to 5 and 7 in terms of generator steps. The temperament, aside from what is on the normal list, tempers out 126/125 and 225/224, and 31edo is a good tuning for it.

Commas: 81/80, 126/125

7 and 9-limit minimax

[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |-3 0 5/2 0>]

Eigenmonzos: 2, 5

valid range: [694.737, 700.000] (19 to 12)

nice range: [694.786, 701.955]

strict range: [694.786, 700.000]

POTE generator: 696.495

Mapping generator: ~3

Algebraic generator: Cybozem, the real root of 15x^3-10x^2-18, which comes to 503.4257 cents. The recurrence converges quickly.

Map: [<1 0 -4 -13|, <0 1 4 10|]

Generators: 2, 3

Wedgie: <<1 4 10 4 13 12||

EDOs: 12, 19, 31, 43, 50, 74, 81, 105, 143b

## Bimeantone

11/8 is mapped to half octave minus the meantone diesis.

Commas: 81/80, 126/125, 245/242

POTE generator: ~3/2 = 696.016

Map: [<2 0 -8 -26 -31|, <0 1 4 10 12|]

EDOs: 12, 38d, 50

### 13-limit

Commas: 81/80, 105/104, 126/125, 245/242

POTE generator: ~3/2 = 695.836

Map: [<2 0 -8 -26 -31 -40|, <0 1 4 10 12 15|]

EDOs: 12f, 50

## Unidecimal meantone aka Huygens

Commas: 81/80, 126/125, 99/98

11-limit minimax

[|1 0 0 0 0>, |25/16 -1/8 0 0 1/16>, |9/4 -1/2 0 0 1/4>, |21/8 -5/4 0 0 5/8>, |25/8 -9/4 0 0 9/8>]

Eigenmonzos: 2, 11/9

valid range: [696.774, 700.000] (31 to 12)

nice range: [691.202, 701.955]

strict range: [696.774, 700.000]

POTE generator: ~3/2 = 696.967

Mapping generator: ~3

Algebraic generator: Traverse, the positive real root of x^4+2x-13, or 696.9529 cents.

Map: [<1 0 -4 -13 -25|, <0 1 4 10 18|]

Generators: 2, 3

EDOs: 12, 31, 43, 74, 105, 198be

### Tridecimal meantone

Commas: 66/65, 81/80, 99/98, 105/104

POTE generator: ~3/2 = 696.642

Mapping generator: ~3

Map: [<1 0 -4 -13 -25 -20|, <0 1 4 10 18 15|]

EDOs: 12f, 31, 43f

### Grosstone

Commas: 81/80, 99/98, 126/125, 144/143

valid range: [696.774, 700.000] (31 to 12)

nice range: [691.202, 701.955]

strict range: [696.774, 700.000]

POTE generator: ~3/2 = 697.264

Mapping generator: ~3

Map: [<1 0 -4 -13 -25 29|, <0 1 4 10 18 -16|]

EDOs: 12, 31, 43, 74, 105

### Meridetone

Commas: 78/77, 81/80, 99/98, 126/125

POTE generator: ~3/2 = 697.529

Mapping generator: ~3

Map: [<1 0 -4 -13 -25 -39|, <0 1 4 10 18 27|]

EDOs: 12f, 31f, 43

### Hemimeantone

Commas: 81/80, 99/98, 126/125, 169/168

POTE generator: ~52/45 = 250.304

Mapping generator: ~26/15

Map: [<1 0 -4 -13 -25 -5|, <0 2 8 20 36 11|]

EDOs: 19e, 43, 62, 167bef

## Meanpop

Commas: 81/80, 126/125, 385/384

11-limit minimax 1/4 comma

[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |-3 0 5/2 0 0>, |11 0 -13/4 0 0>]

Eigenmonzos: 2, 5

valid range: [694.737, 696.774] (19 to 31)

nice range: [691.202, 701.955]

strict range: [694.737, 696.774]

POTE generator: 696.434

Mapping generator: ~3

Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x^3+6x-19. Unlike Cybozem, the recurrence for Radieubiz does not converge.

Map: [<1 0 -4 -13 24|, <0 1 4 10 -13|]

Generators: 2, 3

EDOs: 12e, 19, 31, 50, 81

### 13-limit Meanpop

Commas: 81/80, 105/104, 126/125, 144/143

valid range: [694.737, 696.774] (19 to 31)

nice range: [691.202, 701.955]

strict range: [694.737, 696.774]

POTE generator: ~3/2 = 696.211

Mapping generator: ~3

Map: [<1 0 -4 -13 24 -20|, <0 1 4 10 -13 15|]

EDOS: 12ef, 19, 31, 50, 81

### Meanplop

Commas: 65/64, 78/77, 81/80, 91/90

POTE generator: ~3/2 = 696.202

Mapping generator: ~3

Map: [<1 0 -4 -13 24 10|, <0 1 4 10 -13 -4|]

EDOs: 12e, 19, 31f, 50ff

Commas: 45/44, 56/55, 81/80

POTE generator: ~3/2 = 696.250

Mapping generator: ~3

Map: [<1 0 -4 -13 -6|, <0 1 4 10 6|]

EDOs: 7d, 12, 19, 31e, 50ee

### 13-limit

Commas: 45/44, 56/55, 78/77, 81/80

POTE generator: ~3/2 = 696.146

Mapping generator: ~3

Map: [<1 0 -4 -13 -6 -20|, <0 1 4 10 6 15|]

EDOs: 12f, 19, 31e, 50ee

## Vincenzo

Commas: 45/44, 56/55, 65/64, 81/80

POTE generator: ~3/2 = 695.060

Mapping generator: ~3

Map: [<1 0 -4 -13 -6 10|, <0 1 4 10 6 -4|]

EDOs: 7d, 12, 19, 26d

### 17-limit

Commas: 45/44, 52/51, 56/55, 65/64, 81/80

POTE generator: ~3/2 = 695.858

Map: [<1 0 -4 -13 -6 10 12|, <0 1 4 10 6 -4 -5|]

EDOs: 7d, 12, 19

### 19-limit

Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80

POTE generator: ~3/2 = 696.131

Map: [<1 0 -4 -13 -6 10 12 9|, <0 1 4 10 6 -4 -5 -3|]

EDOs: 7d, 12, 19

### 23-limit

Commas: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80

POTE generator: ~3/2 = 696.044

Map: [<1 0 -4 -13 -6 10 12 9 14|, <0 1 4 10 6 -4 -5 -3 -6|]

EDOs: 7d, 12, 19

### 29-limit

Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80

POTE generator: ~3/2 = 695.913

Map: [<1 0 -4 -13 -6 10 12 9 14 8|, <0 1 4 10 6 -4 -5 -3 -6 -2|]

EDOs: 7d, 12, 19

### 31-limit

Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92

POTE generator: ~3/2 = 695.750

Map: [<1 0 -4 -13 -6 10 12 9 14 8 16|, <0 1 4 10 6 -4 -5 -3 -6 -2 -7|]

EDOs: 7d, 12, 19

### 37-limit

Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92

POTE generator: ~3/2 = 695.603

Map: [<1 0 -4 -13 -6 10 12 9 14 8 16 -9|, <0 1 4 10 6 -4 -5 -3 -6 -2 -7 9|]

EDOs: 7d, 12, 19

### 41-limit

Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123

POTE generator: ~3/2 = 695.696

Map: [<1 0 -4 -13 -6 10 12 9 14 8 16 -9 18|, <0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8|]

EDOs: 7d, 12, 19

### 43-limit

Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123

POTE generator: ~3/2 = 695.688

Map: [<1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7|, <0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1|]

EDOs: 7d, 12, 19

### 47-limit

Commas: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123

POTE generator: ~3/2 = 695.676

Map: [<1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4|, <0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1|]

EDOs: 7d, 12, 19

## Meanundeci

Commas: 33/32, 55/54, 77/75

POTE generator: ~3/2 = 694.689

Mapping generator: ~3

Map: [<1 0 -4 -13 5|, <0 1 4 10 -1|]

EDOs: 5d, 7d, 12e, 19e

### 13-limit

Commas: 33/32, 55/54, 65/64, 77/75

POTE generator: ~3/2 = 694.764

Mapping generator: ~3

Map: [<1 0 -4 -13 5 10|, <0 1 4 10 -1 -4|]

EDOs: 7d, 12e, 19e

## Meanundec

Commas: 27/26, 40/39, 45/44, 56/55

POTE generator: ~3/2 = 697.254

Mapping generator: ~3

Map: [<1 0 -4 -13 -6 -1|, <0 1 4 10 6 3|]

EDOS: 7d, 12f, 19f, 31eff

# Flattone

Commas: 81/80, 525/512

The wedgie for flattone is <<1 4 -9 4 -17 -32||, which tells us among other things that 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished seventh interval. Other intervals are 7/6, a diminished third, and 7/5, a doubly diminshed fifth. Good tunings for flattone are 26edo, 45edo and 64edo.

7-limit minimax

[|1 0 0 0>, |21/13 0 1/13 -1/13>, |32/13 0 4/13 -4/13>, |32/13 0 -9/13 9/13>]

Eigenmonzos: 2, 7/5

9-limit minimax

[|1 0 0 0>, |17/11 2/11 0 -1/11>, |24/11 8/11 0 -4/11>, |34/11 -18/11 0 9/11>]

Eigenmonzos: 2, 9/7

valid range: [692.308, 694.737] (26 to 19)

nice range: [692.353, 701.955]

strict range: [692.353, 694.737]

POTE generator: 693.779

Mapping generator: ~3

Algebraic generator: Squarto, the positive root of 8x^2-4x-9, at 506.3239 cents, equal to (1+sqrt(19))/4.

Map: [<1 0 -4 17|, <0 1 4 -9|]

Wedgie: <<1 4 -9 4 -17 -32||

Generators: 2, 3

EDOs: 7, 19, 26, 45

## 11-limit

Commas: 45/44, 81/80, 385/384

valid range: [692.308, 694.737] (26 to 19)

nice range: [682.502, 701.955]

strict range: [692.308, 694.737]

POTE generator: ~3/2 = 693.126

Mapping generator: ~3

Map: [<1 0 -4 17 -6|, <0 1 4 -9 6|]

EDOs: 7, 19, 26, 45, 71bc, 116bcde

## 13-limit

45/44, 65/64, 78/77, 81/80

valid range: [692.308, 694.737] (26 to 19)

nice range: [682.502, 701.955]

strict range: [692.308, 694.737]

POTE generator: ~3/2 = 693.058

Mapping generator: ~3

Map: [<1 0 -4 17 -6 10|, <0 1 4 -9 6 -4|]

EDOs: 7, 19, 26, 45f, 71bcf, 116bcdef

# Dominant

Commas: 36/35, 64/63

The wedgie for dominant is <<1 4 -2 4 -6 -16||. Now the interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is 12edo, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with 29edo, 41edo, or 53edo.

valid range: [700.000, 720.000] (12 to 5)

nice range: [694.786, 715.587]

strict range: [700.000, 715.587]

POTE generator: 701.573

Mapping generator: ~3

Map: [<1 0 -4 6|, <0 1 4 -2|]

Wedgie: <<1 4 -2 4 -6 -16||

EDOs: 5, 7, 12, 17c, 29cd

## 11-limit

Commas: 36/35, 64/63, 56/55

valid range: [700.000, 705.882] (12 to 17)

nice range: [691.202, 715.587]

strict range: [700.000, 705.882]

POTE generator: ~3/2 = 703.254

Mapping generator: ~3

Map: [<1 0 -4 6 13|, <0 1 4 -2 -6|]

EDOs: 5, 12, 17c, 29cde

## 13-limit

Commas: 36/35, 56/55, 64/63, 66/65

valid range: 705.882 (17)

nice range: [691.202, 715.587]

strict range:705.882

POTE generator: ~3/2 = 703.636

Map: [<1 0 -4 6 13 18|, <0 1 4 -2 -6 -9|]

EDOs: 12f, 17c, 29cdef

## Dominion

Commas: 26/25, 36/35, 56/55, 64/63

POTE generator: ~3/2 = 704.905

Map: [<1 0 -4 6 13 -9|, <0 1 4 -2 -6 8|]

EDOs: 5, 12, 17c, 46cde

## Domineering

Commas: 36/35, 45/44, 64/63

POTE generator: ~3/2 = 698.776

Mapping generator: ~3

Map: [<1 0 -4 6 -6|, <0 1 4 -2 6|]

EDOs: 5e, 7, 12, 19d, 43de

### 13-limit

Commas: 36/35, 45/44, 52/49, 64/63

POTE generator: ~3/2 = 695.762

Mapping generator: ~3

Map: [<1 0 -4 6 -6 10|, <0 1 4 -2 6 -4|]

EDOs: 5ef, 7, 12, 19d, 31def

### 17-limit

Commas: 36/35, 45/44, 51/49, 52/49, 64/63

POTE generator: ~3/2 = 696.115

Mapping generator: ~3

Map: [<1 0 -4 6 -6 10 12|, <0 1 4 -2 6 -4 -5|]

EDOs: 5ef, 7, 12, 19d, 31def

### 19-limit

Commas: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56

POTE generator: ~3/2 = 696.217

Mapping generator: ~3

Map: [<1 0 -4 6 -6 10 12 9|, <0 1 4 -2 6 -4 -5 -3|]

EDOs: 5ef, 7, 12, 19d, 31def

### Dominatrix

Commas: 27/26, 36/35, 45/44, 64/63

POTE generator: ~3/2 = 698.544

Mapping generator: ~3

Map: [<1 0 -4 6 -6 -1|, <0 1 4 -2 6 3|]

EDOs: 5e, 7, 12f, 19df

## Domination

Commas: 36/35, 64/63, 77/75

POTE generator: ~3/2 = 705.004

Mapping generator: ~3

Map: [<1 0 -4 6 -14|, <0 1 4 -2 11|]

EDOs: 5e, 12e, 17c, 46cd

### 13-limit

Commas: 26/25, 36/35, 64/63, 66/65

POTE generator: ~3/2 = 705.496

Mapping generator: ~3

Map: [<1 0 -4 6 -14 -9|, <0 1 4 -2 11 8|]

EDOs: 5e, 12e, 17c

## Arnold

Commas: 22/21, 33/32, 36/35

POTE generator: ~3/2 = 698.491

Mapping generator: ~3

Map: [<1 0 -4 6 5|, <0 1 4 -2 -1|]

EDOs: 5, 7, 12e

### 13-limit

Commas: 22/21, 27/26, 33/32, 36/35

POTE generator: ~3/2 = 696.743

Map: [<1 0 -4 6 5 -1|, <0 1 4 -2 -1 3|]

EDOs: 5, 7, 12ef, 19def

### 17-limit

Commas: 22/21, 27/26, 33/32, 36/35, 51/49

POTE generator: ~3/2 = 696.978

Map: [<1 0 -4 6 5 -1 12|, <0 1 4 -2 -1 3 -5|]

EDOs: 5, 7, 12ef, 19def

### 19-limit

Commas: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56

POTE generator: ~3/2 = 697.068

Map: [<1 0 -4 6 5 -1 12 9|, <0 1 4 -2 -1 3 -5 -3|]

EDOs: 5, 7, 12ef, 19def

# Sharptone

Commas: 21/20, 28/27

Sharptone, with a wedgie <<1 4 3 4 2 -4||, is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. 12edo tuning does sharptone about as well as such a thing can be done, of course not in its patent val.

POTE generator: ~3/2 = 700.140

Mapping generator: ~3

Map: [<1 0 -4 -2|, <0 1 4 3|]

Wedgie: <<1 4 3 4 2 -4||

EDOs: 5, 7d, 12d

## Meanertone

Commas: 21/20, 28/27, 33/32

POTE generator: ~3/2 = 696.615

Map: [<1 0 -4 -2 5|, <0 1 4 3 -1|]

EDOs: 5, 7d, 12de

# Meansept

Commas: 15/14, 81/80

POTE generator: ~3/2 = 682.895

Mapping generator: ~3

Map: [<1 0 -4 -5|, <0 1 4 5|]

Wedgie: <<1 4 5 4 5 0||

EDOs: 5d, 7, 12dd

## 11-limit

Commas: 15/14, 22/21, 81/80

POTE generator: ~3/2 = 685.234

Mapping generator: ~3

Map: [<1 0 -4 -5 -6|, <0 1 4 5 6|]

EDOs: 5de, 7, 12dd

# Supermean

Commas: 81/80, 672/625

POTE generator: ~3/2 = 704.889

Map: [<1 0 -4 -21|, <0 1 4 15|]

EDOs: 5d, 12d, 17c, 29c

## 11-limit

Commas: 56/55, 81/80, 132/125

POTE generator: ~3/2 = 705.096

Map: [<1 0 -4 -21 -14|, <0 1 4 15 11|]

EDOs: 5de, 12de, 17c, 29c

## 13-limit

Commas: 26/25, 56/55, 66/65, 81/80

POTE generator: ~3/2 = 705.094

Map: [<1 0 -4 -21 -14 -9|, <0 1 4 15 11 8|]

EDOs: 5de, 12de, 17c, 29c

# Injera

Commas: 50/49, 81/80

The wedgie for injera is <<2 8 8 8 7 -4||, which tells us it has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. 38edo, which is two parallel 19edos, is an excellent tuning for injera.

valid range: [685.714, 700.000] (14c to 12)

nice range: [688.957, 701.955]

strict range: [688.957, 700.000]

POTE generator: 694.375

Mapping generator: ~3

Map: [<2 0 -8 -7|, <0 1 4 4|]

Wedgie: <<2 8 8 8 7 -4||

EDOs: 12, 26, 38, 102bcd, 140bccd, 178bbccdd

## 11-limit

Commas: 45/44, 50/49, 81/80

valid range: [685.714, 700.000] (14c to 12)

nice range: [682.458, 701.955]

strict range: [685.714, 700.000]

POTE generator: ~3/2 = 692.840

Mapping generator: ~3

Map: [<2 0 -8 -7 -12|, <0 1 4 4 6|]

EDOs: 12, 14c, 26, 90bce, 116bcce

## 13-limit

Commas: 45/44, 50/49, 78/77, 81/80

valid range: 692.308 (26)

nice range: [682.458, 701.955]

strict range: 692.308 (26)

POTE generator: ~3/2 = 692.673

Mapping generator: ~3

Map: [<2 0 -8 -7 -12 -21|, <0 1 4 4 6 9|]

EDOs: 12f, 14cf, 26, 38e

## Enjera

Commas: 27/26, 40/39, 45/44, 50/49

POTE generator: ~3/2 = 694.121

Mapping generator: ~3

Map: [<2 0 -8 -7 -12 -2|, <0 1 4 4 6 3|]

EDOs: 12f, 14c, 26f, 38eff

## Injerous

Commas: 33/32, 50/49, 55/54

POTE generator: ~3/2 = 690.548

Mapping generator: ~3

Map: [<2 0 -8 -7 10|, <0 1 4 4 -1|]

EDOs: 12e, 14c, 26e, 40cee

## Lahoh

Commas: 50/49, 56/55, 81/77

POTE generator: ~3/2 = 699.001

Mapping generator: ~3

Map: [<2 0 -8 -7 7|, <0 1 4 4 0|]

EDOs: 2cd, 12, 14ce

# Godzilla

Main article: Semaphore and Godzilla

Commas: 49/48, 81/80

Godzilla has wedgie <<2 8 1 8 -4 -20||, and tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. 19edo is the perfect godzilla tuning, so much so that's there's not much point in looking elsewhere. Hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.

valid range: [240.000, 257.143] (5 to 14c)

nice range: [231.174, 266.871]

strict range: [240.000, 257.143]

POTE generator: ~8/7 = 252.635

Mapping generator: ~7/4

Map: [<1 0 -4 2|, <0 2 8 1|]

Wedgie: <<2 8 1 8 -4 -20||

EDOs: 5, 9c, 14c, 19, 62d, 81d, 143bd

## 11-limit

Commas: 45/44, 49/48, 81/80

valid range: [252.632, 257.143] (19 to 14c)

nice range: [231.174, 266.871]

strict range: [252.632, 257.143]

POTE generator: ~8/7 = 254.027

Mapping generator: ~7/4

Map: [<1 0 -4 2 -6|, <0 2 8 1 12|]

EDOs: 14c, 19, 33cd, 52cd

## 13-limit

Commas: 45/44, 49/48, 78/77, 81/80

valid range: 694.737 (19)

nice range: [621.581, 737.652]

strict range: 694.737

POTE generator: ~8/7 = 253.603

Mapping generator: ~7/4

Map: [<1 0 -4 2 -6 -5|, <0 2 8 1 12 11|]

EDOs: 14cf, 19, 33cdf, 52cdf

## Semafour

Commas: 33/32, 49/48, 55/54

POTE generator: ~8/7 = 254.042

Mapping generator: ~7/4

Map: [<1 0 -4 2 5|, <0 2 8 1 -2|]

EDOs: 5, 14c, 19e, 33cde

## Varan

Commas: 49/48, 77/75, 81/80

POTE generator: ~8/7 = 251.079

Mapping generator: ~7/4

Map: [<1 0 -4 2 -10|, <0 2 8 1 17|]

EDOs: 19e, 24, 43de

### 13-limit

Commas: 49/48, 66/65, 77/75, 81/80

POTE generator: ~8/7 = 251.165

Mapping generator: ~7/4

Map: [<1 0 -4 2 -10 -5|, <0 2 8 1 17 11|]

EDOs: 19e, 24, 43de

## Baragon

Commas: 49/48, 56/55, 81/80

POTE generator: ~8/7 = 251.173

Mapping generator: ~7/4

Map: [<1 0 -4 2 9|, <0 2 8 1 -7|]

EDOs: 19, 24, 43d

## Music

"Change is on the Wind" in Godzilla[9] by Igliashon Jones

# Mohajira

Commas: 81/80, 6144/6125

Mohajira, with wedgie <<2 8 -11 8 -23 -48||, really makes more sense as an 11-limit temperament. It has a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 128/105. Mohajira tempers out 6144/6125, the porwell comma. 31edo makes for an excellent (7-limit) mohajira tuning, with generator 9/31. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.

Mohajira can also be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 11-limit). Within this paradigm, mohajira is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, that maps four 3/2's to 5/1, and that maps the interval one quarter tone flat of 16/9 to 7/4.

7 and 9-limit minimax 1/4 comma

[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |6 0 -11/8 0>]

Eigenmonzos: 2, 5

POTE generator: ~128/105 = 348.415

Mapping generator: ~128/105

Algebraic generator: Mohabis, real root of 3x^3-3x^2-1, 348.6067 cents. Corresponding recurrence converges quickly.

Map: [<1 1 0 6|, <0 2 8 -11|]

Generators: 2, 128/105

Wedgie: <<2 8 -11 8 -23 -48||

EDOs: 7, 24, 31, 38, 55, 69

## 11-limit

Commas: 81/80, 121/120, 176/175

11-limit minimax 1/4 comma

[|1 0 0 0 0>, |1 0 1/4 0 0>, |0 0 1 0 0>, |6 0 -11/8 0 0>, |2 0 5/8 0 0>]

Eigenmonzos: 2, 5

POTE generator: ~11/9 = 348.477

Mapping generator: ~11/9

Map: [<1 1 0 6 2|, <0 2 8 -11 5|]

Generators: 2, 11/9

EDOs: 7, 24, 31, 38, 55

## 13-limit

Commas: 66/65, 81/80, 105/104, 121/120

POTE generator: ~11/9 = 348.558

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4|, <0 2 8 -11 5 -1|]

EDOs: 7, 24, 31, 38, 55

## 17-limit

Commas: 66/65, 81/80, 105/104, 121/120, 154/153

POTE generator: ~11/9 = 348.736

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4 7|, <0 2 8 -11 5 -1 -10|]

EDOs: 7, 24, 31, 38g, 55

## 19-limit

Commas: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152

POTE generator: ~11/9 = 348.810

Mapping generator: ~11/9

Map: [<1 1 0 6 2 4 7 6|, <0 2 8 -11 5 -1 -10 -6|]

EDOs: 7, 24, 31, 38gh, 55

# Ptolemy

Commas: 81/80, 121/120, 525/512

POTE generator: ~11/9 = 346.922

Map: [<1 1 0 8 2|, <0 2 8 -18 5|]

EDOs: 7, 31dd, 38d, 45e, 83bcddee

## 13-limit

Commas: 65/64, 81/80, 105/104, 121/120

POTE generator: ~11/9 = 346.910

Map: [<1 1 0 8 2 6|, <0 2 8 -18 5 -8|]

EDOs: 7, 31ddf, 38df, 45ef, 83bcddeeff

# Maqamic

Main article: Maqamic

Commas: 81/80, 36/35, 121/120

Maqamic temperament is much like Mohajira, except in that it 36/35 vanishes instead of 176/175. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.

POTE generator: ~11/9 = 350.934

Mapping generator: ~11/9

Map: [<1 1 0 4 2|, <0 2 8 -4 5|]

Generators: 2, 11/9

EDOs: 7, 10c, 17c, 24d, 31d

## 13-limit

Commas: 81/80, 36/35, 121/120, 144/143

POTE generator: ~11/9 = 350.816

Mapping generator: ~11/9

Map: [<1 1 0 4 2 4|, <0 2 8 -4 5 -1|]

Generators: 2, 11/9

EDOs: 7, 10c, 17c, 24d, 31d

# Migration

Commas: 81/80, 121/120, 126/125

POTE generator: ~11/9 = 348.182

Mapping generator: ~11/9

Map: [<1 1 0 -3 2|, <0 2 8 20 5|]

EDOs: 7d, 31, 100de, 131bdee, 162bdee

## 13-limit

Commas: 66/65, 81/80, 121/120, 126/125

POTE generator: ~11/9 = 348.490

Map: [<1 1 0 -3 2 4|, <0 2 8 20 5 -1|]

EDOs: 7d, 24d, 31, 55d

# Mohamaq

Commas: 81/80, 392/375

POTE generator: ~25/21 = 350.586

Mapping generator: ~25/21

Map: [<1 1 0 -1|, <0 2 8 13|]

EDOs: 17c, 24, 65c, 89cd

## 11-limit

Commas: 56/55, 77/75, 243/242

POTE generator: ~11/9 = 350.565

Mapping generator: ~11/9

Map: [<1 1 0 -1 2|, <0 2 8 13 5|]

EDOs: 17c, 24, 65c, 89cd

## 13-limit

Commas: 56/55, 66/65, 77/75, 243/242

POTE generator: ~11/9 = 350.745

Mapping generator: ~11/9

Map: [<1 1 0 -1 2 4|, <0 2 8 13 5 -1|]

EDOs: 17c, 24, 41c, 65c

# Orphic

Commas: 81/80, 5898240/5764801

POTE generator: ~7/6 = 275.794

Mapping generator: ~343/288

Map: [<2 1 -4 4|, <0 4 16 3|]

Wedgie: <<8 32 6 32 -13 -76||

EDOs: 26, 74, 174bd, 248bd

## 11-limit

Commas: 81/80, 99/98, 73728/73205

POTE generator: ~7/6 = 275.762

Mapping generator: ~77/64

Map: [<2 1 -4 4 8|, <0 4 16 3 -2|]

EDOs: 26, 48c, 74, 248bd, 322bd

## 13-limit

Commas: 81/80, 99/98, 144/143, 2200/2197

POTE generator: ~7/6 = 275.774

Mapping generator: ~63/52

Map: [<2 1 -4 4 8 2|, <0 4 16 3 -2 10|]

EDOs: 26, 48c, 74, 174bd, 248bd, 322bd

# Mothra

Commas: 81/80, 1029/1024

Mothra, with wedgie <<3 12 -1 12 -10 -36||, splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using 31edo with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to slendric.

Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.

7 and 9-limit minimax 1/4 comma

[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3 0 -1/12 0>]

Eigenmonzos: 2, 5

POTE generator: ~8/7 = 232.193

Mapping generator: ~8/7

Algebraic generator: Rabrindanath, largest real root of x^8-3x^2+1, or 232.0774 cents.

Map: [<1 1 0 3|, <0 3 12 -1|]

Generators: 2, 8/7

Wedgie: <<3 12 -1 12 -10 -36||

EDOs: 5, 26, 31, 57, 88

## 11-limit

Commas: 81/80, 99/98, 385/384

POTE generator: ~8/7 = 232.031

Mapping generator: ~8/7

Map: [<1 1 0 3 5|, <0 3 12 -1 -8|]

EDOs: 5, 26, 31, 57, 88, 150be, 181bee

## 13-limit

Commas: 81/80, 99/98, 105/104, 144/143

POTE generator: ~8/7 = 231.811

Mapping generator: ~8/7

Map: [<1 1 0 3 5 1|, <0 3 12 -1 -8 14|]

EDOs: 5, 26, 31, 57, 88

## Cynder

Commas: 45/44, 81/80, 1029/1024

POTE generator: ~8/7 = 231.317

Mapping generator: ~8/7

Map: [<1 1 0 3 0|, <0 3 12 -1 18|]

EDOs: 5e, 26, 31e, 57e, 83bce

### 13-limit

Commas: 45/44, 78/77, 81/80, 640/637

POTE generator: ~8/7 = 231.293

Mapping generator: ~8/7

Map: [<1 1 0 3 0 1|, <0 3 12 -1 18 14|]

EDOs: 5e, 26, 31e, 57e, 83bce

## Mosura

Commas: 81/80, 176/175, 540/539

POTE generator: ~8/7 = 232.419

Mapping generator: ~8/7

Map: [<1 1 0 3 -1|, <0 3 12 -1 23|]

EDOs: 31, 36, 67, 98, 129, 160be, 191bce, 222bce, 253bcee

### 13-limit

Commas: 81/80, 144/143, 176/175, 196/195

POTE generator: ~8/7 = 232.640

Mapping generator: ~8/7

Map: [<1 1 0 3 -1 7|, <0 3 12 -1 23 -17|]

EDOs: 31, 36, 67, 98

# Squares

Commas: 81/80, 2401/2400

Squares, with wedgie <<4 16 9 16 3 -24||, splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. 31edo, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.

7 and 9 limit minimax 1/4 comma

[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |3/2 0 9/16 0>]

Eigenmonzos: 2, 5

POTE generator: ~9/7 = 425.942

Mapping generator: ~9/7

Algebraic generator: Sceptre2, the positive root of 9x^2+x-16, or (sqrt(577)-1)/18, which is 425.9311 cents.

Map: [<1 3 8 6|, <0 -4 -16 -9|]

Generators: 2, 9/7

EDOs: 14c, 17c, 31, 45, 76

Music:

## 11-limit

Commas: 81/80, 99/98, 121/120

POTE generator: ~9/7 = 425.957

Mapping generator: ~9/7

Map: [<1 3 8 6 7|, <0 -4 -16 -9 -10|]

EDOs: 14c, 17c, 31, 45e, 76e

## 13-limit

Commas: 66/65, 81/80, 99/98, 121/120

POTE generator: ~9/7 = 425.550

Mapping generator: ~9/7

Map: [<1 3 8 6 7 3|, <0 -4 -16 -9 -10 2|]

EDOs: 14c, 17c, 31, 45e, 79cf

## Agora

Commas: 81/80, 99/98, 105/104, 121/120

POTE generator: ~9/7 = 426.276

Mapping generator: ~9/7

Map: [<1 3 8 6 7 14|, <0 -4 -16 -9 -10 -29|]

EDOs: 14cf, 31, 45ef, 76e

### 17-limit

Commas: 81/80, 99/98, 105/104, 120/119, 121/119

POTE generator: ~9/7 = 426.187

Mapping generator: ~9/7

Map: [<1 3 8 6 7 14 8|, <0 -4 -16 -9 -10 -29 -11|]

EDOs: 14cf, 31, 45ef, 76e

### 19-limit

Commas: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119

POTE generator: ~9/7 = 426.225

Mapping generator: ~9/7

Map: [<1 3 8 6 7 14 8 11|, <0 -4 -16 -9 -10 -29 -11 -19|]

EDOs: 14cf, 31, 45ef, 76e

# Cuboctahedra

Commas: 81/80, 385/384, 1375/1372

POTE generator: ~9/7 = 425.993

Mapping generator: ~9/7

Map: [<1 3 8 6 -4|, <0 -4 -16 -9 21|]

EDOs: 14ce, 17ce, 31, 45, 76, 107b

# Liese

Commas: 81/80, 686/675

Liese, with wedgie <<3 12 11 12 9 -8||, splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. 74edo makes for a good liese tuning, though 19edo can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.

7 and 9 limit minimax 1/4 comma

[|1 0 0 0>, |1 0 1/4 0>, |0 0 1 0>, |2/3 0 11/12 0>]

Eigenmonzos: 2, 5

POTE generator: ~10/7 = 632.406

Mapping generator: ~10/7

Algebraic generator: Radix, the real root of x^5-2x^4+2x^3-2x^2+2x-2, also a root of x^6-x^5-2. The recurrence converges.

Map: [<1 0 -4 -3|, <0 3 12 11|]

Generators: 2, 10/7

EDOs: 17c, 19, 36, 55, 74d

## Liesel

Commas: 56/55, 81/80, 540/539

POTE generator: ~10/7 = 633.073

Mapping generator: ~10/7

Map: [<1 0 -4 -3 4|, <0 3 12 11 -1|]

EDOs: 17c, 19, 36, 55e, 91cee

### 13-limit

Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.

Commas: 56/55, 78/77, 81/80, 91/90

POTE generator: ~10/7 = ~13/9 = 633.042

Mapping generator: ~10/7

Map: [<1 0 -4 -3 4 0|, <0 3 12 11 -1 7|]

EDOs: 17c, 19, 36, 55ef, 91ceef

## Elisa

Commas: 77/75, 81/80, 99/98

POTE generator: ~10/7 = 633.061

Mapping generator: ~10/7

Map: [<1 0 -4 -3 -5|, <0 3 12 11 16|]

EDOs: 17c, 19e, 36e

## Lisa

Commas: 45/44, 81/80, 343/330

POTE generator: ~10/7 = 631.370

Mapping generator: ~10/7

Map: [<1 0 -4 -3 -6|, <0 3 12 11 18|]

EDOs: 19

### 13-limit

Commas: 45/44, 81/80, 91/88, 147/143

POTE generator: ~10/7 = 631.221

Map: [<1 0 -4 -3 -6 0|, <0 3 12 11 18 7|]

EDOs: 19

# Jerome

Jerome is related to Hieronymus' tuning; the Hieronymus generator is 5^(1/20), or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.

Commas: 81/80, 17280/16807

POTE generator: ~54/49 = 139.343

Mapping generator: ~54/49

Map: [<1 1 0 2|, <0 5 20 7|]

Wedgie: <<5 30 7 20 -3 -40||

EDOs: 9c, 17c, 26, 43, 69, 112bd

## 11-limit

Commas: 81/80, 99/98, 864/847

POTE generator: ~12/11 = 139.428

Mapping generator: ~12/11

Map: [<1 1 0 2 3|, <0 5 20 7 4|]

EDOs: 9c, 17c, 26, 43, 69

## 13-limit

Commas: 78/77, 81/80, 99/98, 144/143

POTE generator: ~13/12 = 139.387

Mapping generator: ~12/11

Map: [<1 1 0 2 3 3|, <0 5 20 7 4 6|]

EDOs: 9c, 17c, 26, 43, 69

## 17-limit

Commas: 78/77, 81/80, 99/98, 144/143, 189/187

POTE generator: ~13/12 = 139.362

Mapping generator: ~12/11

Map: [<1 1 0 2 3 3 2|, <0 5 20 7 4 6 18|]

EDOs: 26, 43, 69

## 19-limit

Commas: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143

POTE generator: ~13/12 = 139.313

Mapping generator: ~12/11

Map: [<1 1 0 2 3 3 2 1|, <0 5 20 7 4 6 18 28|]

EDOs: 26, 43, 69

# Meanmag

Commas: 81/80, 3125/3072

POTE generator: ~8/7 = 238.396

Mapping generator: ~7

Map: [<19 30 44 0|, <0 0 0 1|]

Wedgie: <<0 0 19 0 30 44||

EDOs: 19, 38, 57, 76, 95bc

# Undevigintone

Commas: 49/48, 81/80, 126/125

POTE generator: ~11/8 = 538.047

Mapping generator: ~11

Map: [<19 30 44 53 0|, <0 0 0 0 1|]

EDOs: 19, 38d